Fast Direct Methods for Gaussian Processes and the Analysis of NASA Kepler Mission Data

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Abstract

A number of problems in probability and statistics can be addressed using the multivariate normal (or multivariate Gaussian) distribution. In the one-dimensional case, computing the probability for a given mean and variance simply requires the evaluation of the corresponding Gaussian density. In the n-dimensional setting, however, it requires the inversion of an n×n covariance matrix, C, as well as the evaluation of its determinant, det(C). In many cases, the covariance matrix is of the form C=σ2I+K, where K is computed using a specified kernel, which depends on the data and additional parameters (called hyperparameters in Gaussian process computations). The matrix C is typically dense, causing standard direct methods for inversion and determinant evaluation to require O(n3) work. This cost is prohibitive for large-scale modeling. Here, we show that for the most commonly used covariance functions, the matrix C can be hierarchically factored into a product of block low-rank updates of the identity matrix, yielding an O(nlog2n) algorithm for inversion, as discussed in Ambikasaran and Darve, 2013. More importantly, we show that this factorization enables the evaluation of the determinant det(C), permitting the direct calculation of probabilities in high dimensions under fairly broad assumption about the kernel defining K. Our fast algorithm brings many problems in marginalization and the adaptation of hyperparameters within practical reach using a single CPU core. The combination of nearly optimal scaling in terms of problem size with high-performance computing resources will permit the modeling of previously intractable problems. We illustrate the performance of the scheme on standard covariance kernels, and apply it to a real data set obtained from the Kepler Mission.

Author

Sivaram Ambikasaran, Daniel Foreman-Mackey, Leslie Greengard, David W. Hogg, Michael O’Neil

Journal

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Paper Publication Date

2014

Paper Type

Astrostatistics